Stress-energy-momentum pseudotensor

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In the theory of general relativity, the stress-energy-momentum pseudotensor or Landau-Lifshitz pseudotensor allows the energy-momentum of a system of gravitating matter to be defined; in particular it allows the total matter plus the gravitating energy-momentum to form a conserved current within the framework of general relativity, so that the total energy-momentum crossing the hypersurface of any closed space-time hypervolume vanishes.

The use of the Landau-Lifshitz combined matter+gravitational stress-energy-momentum pseudotensor^[1] allows the energy-momentum conservation laws to be extended into general relativity. Subtraction of the matter stress-energy-momentum tensor from the combined pseudotensor results in the gravitational stress-energy-momentum pseudotensor. Some people object to this derivation on the grounds that pseudotensors are inappropriate objects in general relativity, but this treatment only requires the use of the 4-divergence of a pseudotensor which is, in this case, a tensor. On the other hand, most pseudotensors are sections of jet bundles, which are perfectly valid objects in GR.

[edit] Definition and properties of the Landau-Lifshitz pseudotensor

Landau & Lifshitz were led by four requirements in their search for a gravitational energy momentum pseudotensor, $t_{LL}^{\mu \nu}$ :^[1]

that it be constructed entirely from the metric tensor, so as to be purely geometrical or gravitational in origin.
that it be index symmetric , i.e. $t_{LL}^{\mu \nu} = t_{LL}^{\nu \mu}$ , (to conserve angular momentum)
that, when added to the stress-energy tensor of matter, $T μν$ , its total 4-divergence vanishes (this is required of any conserved current) so that we have a conserved expression for the total stress-energy-momentum.
that it vanish locally in an inertial frame of reference (which requires that it only contains first and not second derivatives of the metric), as might be expected by the equivalence principle.

Landau and Lifshitz showed that there is a unique construction that satisfies these requirements, namely

$t_{LL}^{\mu \nu} = - \frac{1}{8\pi G}(G^{\mu \nu}) + \frac{1}{16\pi G (-g)}((-g)(g^{\mu \nu}g^{\alpha \beta} - g^{\mu \alpha}g^{\nu \beta})),_{\alpha \beta}$

where:

$G μν$ is the Einstein tensor (which is constructed from the metric)

$g μν$ is the metric tensor

$g = \, \det \, (g_{\mu \nu})$ is the determinant of a spacetime Lorentz metric

$, αβ$ are partial derivatives, not covariant derivatives.

G is Newton's gravitational constant.

Examining the 4 conditions we can see that the first 3 are relatively easy to demonstrate:

Since the Einstein tensor ( $G μν$ ) is itself constructed from the metric, so therefore is $t_{LL}^{\mu \nu}$
Since the Einstein tensor ( $G μν$ ) is symmetric so is $t_{LL}^{\mu \nu}$ since the additional terms are symmetric by inspection.
The Landau-Lifshitz pseudotensor is constructed so that when added to the stress-energy tensor of matter, $T μν$ , its total 4-divergence vanishes: $((-g)(T^{\mu \nu} + t_{LL}^{\mu \nu})),{\mu} = 0$ . This follows from the cancellation of the Einstein tensor, $G μν$ , with the stress-energy tensor, $T μν$ by the Einstein field equations; the remaining term vanishes algebraically due the commutativity of partial derivatives applied across antisymmetric indices.
The above construction appears to include second derivative terms in the metric, but in fact the explicit second derivative terms in the expression above cancel with terms contained within the Einstein tensor. This is more evident when the pseudotensor is directly expressed in terms of the metric tensor or the Levi-Civita connection. As a result only the first derivative terms in the metric survive and these vanish where the frame is locally inertial. As a result the entire pseudotensor vanishes locally $t_{LL}^{\mu \nu} = 0$ , which demonstrates the delocalisation of gravitational energy-momentum.^[1]